56.32.9 problem Ex 9

Internal problem ID [14267]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 55. Summary. Page 129
Problem number : Ex 9
Date solved : Friday, October 03, 2025 at 07:29:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 n x \left (1+x \right ) y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.092 (sec). Leaf size: 85

ode:=x^2*diff(diff(y(x),x),x)-2*n*x*(1+x)*diff(y(x),x)+(a^2*x^2+n^2+n)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{n x} x^{n} \left (\operatorname {WhittakerM}\left (\frac {i n^{2}}{\sqrt {a -n}\, \sqrt {a +n}}, \frac {1}{2}, 2 i \sqrt {a -n}\, \sqrt {a +n}\, x \right ) c_1 +\operatorname {WhittakerW}\left (\frac {i n^{2}}{\sqrt {a -n}\, \sqrt {a +n}}, \frac {1}{2}, 2 i \sqrt {a -n}\, \sqrt {a +n}\, x \right ) c_2 \right ) \]

✗ Mathematica

ode=x^2*D[y[x],{x,2}]-2*n*x*(1+x)*D[y[x],x]+(n^2+n+a^2*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-2*n*x*(x + 1)*Derivative(y(x), x) + x**2*Derivative(y(x), (x, 2)) + (a**2*x**2 + n**2 + n)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None